Nichiren Daishonin first taught the invocation of Nam Myoho Renge Kyo in 1253AD. The phrase Nam Myoho Renge Kyo probably needs to be translated for most people. Since this phrase really pertains to every aspect of everything, a simple definition would not be complete, but it would certainly be a good starting point.
Most people like to explain Nam Myoho Renge Kyo from the beginning. I find that starting from the middle provides a more succinct train of thought.
..of Nam Myoho Renge Kyo literally means "cause and effect"
Cause and effect is much like the principle you were taught in science class. Every time you do something, some other appropriate event follows.
For instance, if you turn the key in the ignition of your car, your car starts. If you fail to make this simple cause, prepare to walk.
This seems simple, but the basic law of cause and effect includes complexities that aren't obvious. Even the simple act of turning the car key, starts a chain of events. The key turns on the ignition and just about everything else in your car. Turning the key further causes power to go to the starter motor which turns the engine over, thus starting the first cycle. Once the engine is turning on its' own, releasing the key stops turning the starter motor (otherwise it grinds). Skipping any one of these events leads to a failure to start the car.
Of course, the real complexity is, that the law of cause and effect involves everything that ever happens in our life.
..of Nam Myoho Renge Kyo literally means "mystic law"
Mystic in this case refers to the fact that we don't always understand the link between a cause and an effect.
With the exception of mechanically inclined people, the details of starting your car tends to be a mystery. If the car doesn't start like it usually does, it becomes so much of a mystery that many people must take it to an expert to get it working again.
I've seen people who have run out of gas ask "what did I do to deserve this?" In this case, the answer isn't really what they did, but what they didn't do. Of course I'm referring to that brief stop at the gas station. Many of our problems are due to a lack of cause.
Another difficulty understanding the law of cause and effect is the amount of time between the cause and the effect. Seeing this connection is sometimes impossible, but it is still there. Cause and effect works whether we understand it or not.
One of the benefits of this practice is understanding the relationship between cause and effect. It seems to bring the cause and the effect closer together so that we can recognize the relationship easier.
..of Nam Myoho Renge Kyo literally means "Sutra"
Sutra's are the teachings of the Buddha. Kyo also refers to sound or rhythm of your voice and the universe.
Since the law of cause and effect can be mysterious at times, we need a method of understanding things in order to have any control over the outcome. The act of chanting, puts us in touch with the law of cause and effect on a more subconscious level.
Myoho Renge Kyo
..of Nam Myoho Renge Kyo literally means "mystic law of cause and effect through the teachings of the Buddha"
At this point, we have explained Myoho Renge Kyo which also happens to be the title of the Lotus Sutra.
The Lotus Sutra was taught by Shakyamuni Buddha at about 300BC. After about 30 years of teaching Buddhism to the people, he began teaching the Lotus Sutra.
Shakyamuni's first order of the day was to ask everyone present to forget everything else he had previously taught, because the Lotus Sutra was the most important teaching of all. The Lotus Sutra required about 8 more years of teaching.
Nichiren Daishonin (1222-1282AD) realized the importance of the Lotus Sutra. He agreed with Shakyamuni that the Lotus Sutra was greatest sutra.
..of Nam Myoho Renge Kyo literally means "devotion to"
If you devote yourself to something, you become like that something. If you devote yourself to playing the piano, you become better at playing the piano. Likewise, if you devote time to fixing cars, you can become the expert who can make other peoples cars start.
In simple terms, you get back what you think about the most. If you devote yourself to Myoho Renge Kyo, you start to understand the principles of Myoho Renge Kyo. Since this really involves all aspects of life, it is the most complete devotion you can have.
Getting it all together
.. Nam Myoho Renge Kyo
When Nichiren first taught about Nam Myoho Renge Kyo in 1253AD, it was truly a revolution of thought.
If you think of the book title "Gone with the Wind," you begin to think of the entire story with all the scenes and intricate details. The title indicates the entire story and by telling someone the title, they are going to think of the entire story as well.
Likewise, by invoking the name of the Lotus Sutra, you are really indicating the whole book. If you devote yourself to Myoho Renge Kyo, you become like the Lotus Sutra. By invoking the phrase "Nam Myoho Renge Kyo" once, you essentially invoke the entire Lotus Sutra.
Nichiren Daishonin stepped outside of the normal beliefs of his day by teaching that Buddhism offers the possibility of happiness and fulfillment in the present life. He also taught that anyone can attain enlightenment, as we are.